3.4.92 \(\int x^5 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [392]

Optimal. Leaf size=147 \[ -\frac {1}{36} b d^3 n x^6-\frac {b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac {3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-1/36*b*d^3*n*x^6-1/9*b*e^3*n*x^(6+3*r)/(2+r)^2-3/4*b*d*e^2*n*x^(6+2*r)/(3+r)^2-3*b*d^2*e*n*x^(6+r)/(6+r)^2+1/
6*(d^3*x^6+2*e^3*x^(6+3*r)/(2+r)+9*d*e^2*x^(6+2*r)/(3+r)+18*d^2*e*x^(6+r)/(6+r))*(a+b*ln(c*x^n))

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Rubi [A]
time = 0.25, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {276, 2371, 12, 14} \begin {gather*} \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{36} b d^3 n x^6-\frac {3 b d^2 e n x^{r+6}}{(r+6)^2}-\frac {3 b d e^2 n x^{2 (r+3)}}{4 (r+3)^2}-\frac {b e^3 n x^{3 (r+2)}}{9 (r+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

-1/36*(b*d^3*n*x^6) - (b*e^3*n*x^(3*(2 + r)))/(9*(2 + r)^2) - (3*b*d*e^2*n*x^(2*(3 + r)))/(4*(3 + r)^2) - (3*b
*d^2*e*n*x^(6 + r))/(6 + r)^2 + ((d^3*x^6 + (2*e^3*x^(3*(2 + r)))/(2 + r) + (9*d*e^2*x^(2*(3 + r)))/(3 + r) +
(18*d^2*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
 &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
 x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2371

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{6} x^5 \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int x^5 \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right ) \, dx\\ &=\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (b n) \int \left (d^3 x^5+\frac {18 d^2 e x^{5+r}}{6+r}+\frac {9 d e^2 x^{5+2 r}}{3+r}+\frac {2 e^3 x^{5+3 r}}{2+r}\right ) \, dx\\ &=-\frac {1}{36} b d^3 n x^6-\frac {b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac {3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 156, normalized size = 1.06 \begin {gather*} \frac {1}{36} x^6 \left (6 b d^3 n \log (x)+d^3 \left (6 a-b n-6 b n \log (x)+6 b \log \left (c x^n\right )\right )+\frac {4 e^3 x^{3 r} \left (-b n+3 a (2+r)+3 b (2+r) \log \left (c x^n\right )\right )}{(2+r)^2}+\frac {27 d e^2 x^{2 r} \left (-b n+2 a (3+r)+2 b (3+r) \log \left (c x^n\right )\right )}{(3+r)^2}+\frac {108 d^2 e x^r \left (-b n+a (6+r)+b (6+r) \log \left (c x^n\right )\right )}{(6+r)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]

[Out]

(x^6*(6*b*d^3*n*Log[x] + d^3*(6*a - b*n - 6*b*n*Log[x] + 6*b*Log[c*x^n]) + (4*e^3*x^(3*r)*(-(b*n) + 3*a*(2 + r
) + 3*b*(2 + r)*Log[c*x^n]))/(2 + r)^2 + (27*d*e^2*x^(2*r)*(-(b*n) + 2*a*(3 + r) + 2*b*(3 + r)*Log[c*x^n]))/(3
 + r)^2 + (108*d^2*e*x^r*(-(b*n) + a*(6 + r) + b*(6 + r)*Log[c*x^n]))/(6 + r)^2))/36

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 4021, normalized size = 27.35

method result size
risch \(\text {Expression too large to display}\) \(4021\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/6*x^6*b*(2*e^3*r^2*(x^r)^3+9*d*e^2*r^2*(x^r)^2+18*e^3*r*(x^r)^3+d^3*r^3+18*d^2*e*r^2*x^r+72*d*e^2*r*(x^r)^2+
36*e^3*(x^r)^3+11*d^3*r^2+90*d^2*e*r*x^r+108*d*e^2*(x^r)^2+36*d^3*r+108*d^2*e*x^r+36*d^3)/(2+r)/(3+r)/(6+r)*ln
(x^n)-1/36*x^6*(-7776*e^3*(x^r)^3*a-23328*d^2*e*x^r*a-23328*d*e^2*(x^r)^2*a+2592*I*Pi*b*d^3*r^3*csgn(I*c*x^n)^
3+6264*I*Pi*b*d^3*r^2*csgn(I*c*x^n)^3+3*I*Pi*b*d^3*r^6*csgn(I*c*x^n)^3+1080*b*d^2*e*n*r^3*x^r-24624*ln(c)*b*d*
e^2*r^2*(x^r)^2-38880*ln(c)*b*d*e^2*r*(x^r)^2-6*a*d^3*r^6-132*a*d^3*r^5-1158*a*d^3*r^4+27*I*Pi*b*d*e^2*r^5*csg
n(I*c*x^n)^3*(x^r)^2+3888*I*Pi*b*d^3*csgn(I*c*x^n)^3+27*I*Pi*b*d*e^2*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(
x^r)^2+54*I*Pi*b*d^2*e*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-66*I*Pi*b*d^3*r^5*csgn(I*x^n)*csgn(I*c*x^n)
^2-7776*a*d^3-7344*a*d*e^2*r^3*(x^r)^2-24624*a*d*e^2*r^2*(x^r)^2-38880*a*d*e^2*r*(x^r)^2-10476*a*d^2*e*r^3*x^r
-30456*a*d^2*e*r^2*x^r-42768*a*d^2*e*r*x^r+b*d^3*n*r^6+22*b*d^3*n*r^5+193*b*d^3*n*r^4-3672*I*Pi*b*d*e^2*r^3*cs
gn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-3672*I*Pi*b*d*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-6*I*Pi*b*e^3*r^5*csg
n(I*c)*csgn(I*c*x^n)^2*(x^r)^3+3996*b*d^2*e*n*r^2*x^r+5184*b*d*e^2*n*r*(x^r)^2+6480*b*d^2*e*n*r*x^r+27*b*d*e^2
*n*r^4*(x^r)^2+432*b*d*e^2*n*r^3*(x^r)^2+108*b*d^2*e*n*r^4*x^r-5184*a*d^3*r^3-12528*a*d^3*r^2-15552*a*d^3*r-6*
ln(c)*b*d^3*r^6-132*ln(c)*b*d^3*r^5-1158*ln(c)*b*d^3*r^4-5184*ln(c)*b*d^3*r^3-12528*ln(c)*b*d^3*r^2-15552*ln(c
)*b*d^3*r+1296*b*d^3*n-12*a*e^3*r^5*(x^r)^3-240*a*e^3*r^4*(x^r)^3-7776*ln(c)*b*e^3*(x^r)^3+1296*b*e^3*n*(x^r)^
3-1836*a*e^3*r^3*(x^r)^3-7776*d^3*b*ln(c)-579*I*Pi*b*d^3*r^4*csgn(I*c)*csgn(I*c*x^n)^2-579*I*Pi*b*d^3*r^4*csgn
(I*x^n)*csgn(I*c*x^n)^2-7776*I*Pi*b*d^3*r*csgn(I*x^n)*csgn(I*c*x^n)^2+864*b*d^3*n*r^3+2088*b*d^3*n*r^2+2592*b*
d^3*n*r+468*b*e^3*n*r^2*(x^r)^3+1296*b*e^3*n*r*(x^r)^3+3888*b*d*e^2*n*(x^r)^2+3888*b*d^2*e*n*x^r-23328*ln(c)*b
*d^2*e*x^r-19440*I*Pi*b*d*e^2*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-19440*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^
n)^2*(x^r)^2-6696*a*e^3*r^2*(x^r)^3-11664*a*e^3*r*(x^r)^3-23328*ln(c)*b*d*e^2*(x^r)^2-1836*ln(c)*b*e^3*r^3*(x^
r)^3-6696*ln(c)*b*e^3*r^2*(x^r)^3-11664*ln(c)*b*e^3*r*(x^r)^3-12*ln(c)*b*e^3*r^5*(x^r)^3-240*ln(c)*b*e^3*r^4*(
x^r)^3+4*b*e^3*n*r^4*(x^r)^3+72*b*e^3*n*r^3*(x^r)^3-54*a*d*e^2*r^5*(x^r)^2-1026*a*d*e^2*r^4*(x^r)^2-108*a*d^2*
e*r^5*x^r-1728*a*d^2*e*r^4*x^r-5238*I*Pi*b*d^2*e*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-15228*I*Pi*b*d^2*e*r^2*cs
gn(I*c)*csgn(I*c*x^n)^2*x^r-15228*I*Pi*b*d^2*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-21384*I*Pi*b*d^2*e*r*csgn(I
*x^n)*csgn(I*c*x^n)^2*x^r+6*I*Pi*b*e^3*r^5*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3-27*I*Pi*b*d*e^2*r^5*csg
n(I*c)*csgn(I*c*x^n)^2*(x^r)^2+918*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+19440*I*Pi*b*d*e
^2*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+21384*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r+
3672*I*Pi*b*d*e^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^2+2376*b*d*e^2*n*r^2*(x^r)^2+7776*I*Pi*b*d^3*r
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*Pi*b*d^3*r^6*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2592*I*Pi*b*d^3*r^3*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3348*I*Pi*b*e^3*r^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3-108*ln(c)*b*d^2*e*r^
5*x^r-513*I*Pi*b*d*e^2*r^4*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-3888*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+66*I*
Pi*b*d^3*r^5*csgn(I*c*x^n)^3+579*I*Pi*b*d^3*r^4*csgn(I*c*x^n)^3+11664*I*Pi*b*d^2*e*csgn(I*c*x^n)^3*x^r+6*I*Pi*
b*e^3*r^5*csgn(I*c*x^n)^3*(x^r)^3+3888*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-3888*I*Pi*b*e^3*csgn(I*x
^n)*csgn(I*c*x^n)^2*(x^r)^3+11664*I*Pi*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+3888*I*Pi*b*e^3*csgn(I*c*x^n)^3*(x^r)^3
+7776*I*Pi*b*d^3*r*csgn(I*c*x^n)^3-21384*I*Pi*b*d^2*e*r*csgn(I*c)*csgn(I*c*x^n)^2*x^r+3348*I*Pi*b*e^3*r^2*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+5238*I*Pi*b*d^2*e*r^3*csgn(I*c*x^n)^3*x^r+12312*I*Pi*b*d*e^2*r^2*csgn(
I*c*x^n)^3*(x^r)^2-5832*I*Pi*b*e^3*r*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+6264*I*Pi*b*d^3*r^2*csgn(I*c)*csgn(I*x^
n)*csgn(I*c*x^n)-11664*I*Pi*b*d*e^2*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^2-11664*I*Pi*b*d^2*e*csgn(I*c)*csgn(I*c*x^
n)^2*x^r-3348*I*Pi*b*e^3*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-2592*I*Pi*b*d^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2
-2592*I*Pi*b*d^3*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1026*ln(c)*b*d*e^2*r^4*(x^r)^2-10476*ln(c)*b*d^2*e*r^3*x^r-30
456*ln(c)*b*d^2*e*r^2*x^r-42768*ln(c)*b*d^2*e*r*x^r-7344*ln(c)*b*d*e^2*r^3*(x^r)^2-120*I*Pi*b*e^3*r^4*csgn(I*x
^n)*csgn(I*c*x^n)^2*(x^r)^3-513*I*Pi*b*d*e^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+5832*I*Pi*b*e^3*r*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)*(x^r)^3+918*I*Pi*b*e^3*r^3*csgn(I*c*x^n)^3*(x^r)^3-3*I*Pi*b*d^3*r^6*csgn(I*c)*csg
n(I*c*x^n)^2-3*I*Pi*b*d^3*r^6*csgn(I*x^n)*csgn(I*c*x^n)^2-918*I*Pi*b*e^3*r^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3
+21384*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r+3348*I*Pi*b*e^3*r^2*csgn(I*c*x^n)^3*(x^r)^3+5832*I*Pi*b*e^3*r*csgn(I
*c*x^n)^3*(x^r)^3-3888*I*Pi*b*e^3*csgn(I*c)*csgn(I*c*x^n)^2*(x^r)^3+513*I*Pi*b*d*e^2*r^4*csgn(I*c)*csgn(I*x^n)
*csgn(I*c*x^n)*(x^r)^2+864*I*Pi*b*d^2*e*r^4*csg...

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Maxima [A]
time = 0.29, size = 218, normalized size = 1.48 \begin {gather*} -\frac {1}{36} \, b d^{3} n x^{6} + \frac {1}{6} \, b d^{3} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{3} x^{6} + \frac {b e^{3} x^{3 \, r + 6} \log \left (c x^{n}\right )}{3 \, {\left (r + 2\right )}} + \frac {3 \, b d e^{2} x^{2 \, r + 6} \log \left (c x^{n}\right )}{2 \, {\left (r + 3\right )}} + \frac {3 \, b d^{2} e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e^{3} n x^{3 \, r + 6}}{9 \, {\left (r + 2\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 6}}{3 \, {\left (r + 2\right )}} - \frac {3 \, b d e^{2} n x^{2 \, r + 6}}{4 \, {\left (r + 3\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 6}}{2 \, {\left (r + 3\right )}} - \frac {3 \, b d^{2} e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 6}}{r + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")

[Out]

-1/36*b*d^3*n*x^6 + 1/6*b*d^3*x^6*log(c*x^n) + 1/6*a*d^3*x^6 + 1/3*b*e^3*x^(3*r + 6)*log(c*x^n)/(r + 2) + 3/2*
b*d*e^2*x^(2*r + 6)*log(c*x^n)/(r + 3) + 3*b*d^2*e*x^(r + 6)*log(c*x^n)/(r + 6) - 1/9*b*e^3*n*x^(3*r + 6)/(r +
 2)^2 + 1/3*a*e^3*x^(3*r + 6)/(r + 2) - 3/4*b*d*e^2*n*x^(2*r + 6)/(r + 3)^2 + 3/2*a*d*e^2*x^(2*r + 6)/(r + 3)
- 3*b*d^2*e*n*x^(r + 6)/(r + 6)^2 + 3*a*d^2*e*x^(r + 6)/(r + 6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (137) = 274\).
time = 0.38, size = 868, normalized size = 5.90 \begin {gather*} \frac {6 \, {\left (b d^{3} r^{6} + 22 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} + 864 \, b d^{3} r^{3} + 2088 \, b d^{3} r^{2} + 2592 \, b d^{3} r + 1296 \, b d^{3}\right )} x^{6} \log \left (c\right ) + 6 \, {\left (b d^{3} n r^{6} + 22 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} + 864 \, b d^{3} n r^{3} + 2088 \, b d^{3} n r^{2} + 2592 \, b d^{3} n r + 1296 \, b d^{3} n\right )} x^{6} \log \left (x\right ) - {\left ({\left (b d^{3} n - 6 \, a d^{3}\right )} r^{6} + 22 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{5} + 1296 \, b d^{3} n + 193 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{4} - 7776 \, a d^{3} + 864 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{3} + 2088 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{2} + 2592 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r\right )} x^{6} + 4 \, {\left (3 \, {\left (b r^{5} + 20 \, b r^{4} + 153 \, b r^{3} + 558 \, b r^{2} + 972 \, b r + 648 \, b\right )} x^{6} e^{3} \log \left (c\right ) + 3 \, {\left (b n r^{5} + 20 \, b n r^{4} + 153 \, b n r^{3} + 558 \, b n r^{2} + 972 \, b n r + 648 \, b n\right )} x^{6} e^{3} \log \left (x\right ) + {\left (3 \, a r^{5} - {\left (b n - 60 \, a\right )} r^{4} - 9 \, {\left (2 \, b n - 51 \, a\right )} r^{3} - 9 \, {\left (13 \, b n - 186 \, a\right )} r^{2} - 324 \, b n - 324 \, {\left (b n - 9 \, a\right )} r + 1944 \, a\right )} x^{6} e^{3}\right )} x^{3 \, r} + 27 \, {\left (2 \, {\left (b d r^{5} + 19 \, b d r^{4} + 136 \, b d r^{3} + 456 \, b d r^{2} + 720 \, b d r + 432 \, b d\right )} x^{6} e^{2} \log \left (c\right ) + 2 \, {\left (b d n r^{5} + 19 \, b d n r^{4} + 136 \, b d n r^{3} + 456 \, b d n r^{2} + 720 \, b d n r + 432 \, b d n\right )} x^{6} e^{2} \log \left (x\right ) + {\left (2 \, a d r^{5} - {\left (b d n - 38 \, a d\right )} r^{4} - 16 \, {\left (b d n - 17 \, a d\right )} r^{3} - 144 \, b d n - 8 \, {\left (11 \, b d n - 114 \, a d\right )} r^{2} + 864 \, a d - 96 \, {\left (2 \, b d n - 15 \, a d\right )} r\right )} x^{6} e^{2}\right )} x^{2 \, r} + 108 \, {\left ({\left (b d^{2} r^{5} + 16 \, b d^{2} r^{4} + 97 \, b d^{2} r^{3} + 282 \, b d^{2} r^{2} + 396 \, b d^{2} r + 216 \, b d^{2}\right )} x^{6} e \log \left (c\right ) + {\left (b d^{2} n r^{5} + 16 \, b d^{2} n r^{4} + 97 \, b d^{2} n r^{3} + 282 \, b d^{2} n r^{2} + 396 \, b d^{2} n r + 216 \, b d^{2} n\right )} x^{6} e \log \left (x\right ) + {\left (a d^{2} r^{5} - {\left (b d^{2} n - 16 \, a d^{2}\right )} r^{4} - 36 \, b d^{2} n - {\left (10 \, b d^{2} n - 97 \, a d^{2}\right )} r^{3} + 216 \, a d^{2} - {\left (37 \, b d^{2} n - 282 \, a d^{2}\right )} r^{2} - 12 \, {\left (5 \, b d^{2} n - 33 \, a d^{2}\right )} r\right )} x^{6} e\right )} x^{r}}{36 \, {\left (r^{6} + 22 \, r^{5} + 193 \, r^{4} + 864 \, r^{3} + 2088 \, r^{2} + 2592 \, r + 1296\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")

[Out]

1/36*(6*(b*d^3*r^6 + 22*b*d^3*r^5 + 193*b*d^3*r^4 + 864*b*d^3*r^3 + 2088*b*d^3*r^2 + 2592*b*d^3*r + 1296*b*d^3
)*x^6*log(c) + 6*(b*d^3*n*r^6 + 22*b*d^3*n*r^5 + 193*b*d^3*n*r^4 + 864*b*d^3*n*r^3 + 2088*b*d^3*n*r^2 + 2592*b
*d^3*n*r + 1296*b*d^3*n)*x^6*log(x) - ((b*d^3*n - 6*a*d^3)*r^6 + 22*(b*d^3*n - 6*a*d^3)*r^5 + 1296*b*d^3*n + 1
93*(b*d^3*n - 6*a*d^3)*r^4 - 7776*a*d^3 + 864*(b*d^3*n - 6*a*d^3)*r^3 + 2088*(b*d^3*n - 6*a*d^3)*r^2 + 2592*(b
*d^3*n - 6*a*d^3)*r)*x^6 + 4*(3*(b*r^5 + 20*b*r^4 + 153*b*r^3 + 558*b*r^2 + 972*b*r + 648*b)*x^6*e^3*log(c) +
3*(b*n*r^5 + 20*b*n*r^4 + 153*b*n*r^3 + 558*b*n*r^2 + 972*b*n*r + 648*b*n)*x^6*e^3*log(x) + (3*a*r^5 - (b*n -
60*a)*r^4 - 9*(2*b*n - 51*a)*r^3 - 9*(13*b*n - 186*a)*r^2 - 324*b*n - 324*(b*n - 9*a)*r + 1944*a)*x^6*e^3)*x^(
3*r) + 27*(2*(b*d*r^5 + 19*b*d*r^4 + 136*b*d*r^3 + 456*b*d*r^2 + 720*b*d*r + 432*b*d)*x^6*e^2*log(c) + 2*(b*d*
n*r^5 + 19*b*d*n*r^4 + 136*b*d*n*r^3 + 456*b*d*n*r^2 + 720*b*d*n*r + 432*b*d*n)*x^6*e^2*log(x) + (2*a*d*r^5 -
(b*d*n - 38*a*d)*r^4 - 16*(b*d*n - 17*a*d)*r^3 - 144*b*d*n - 8*(11*b*d*n - 114*a*d)*r^2 + 864*a*d - 96*(2*b*d*
n - 15*a*d)*r)*x^6*e^2)*x^(2*r) + 108*((b*d^2*r^5 + 16*b*d^2*r^4 + 97*b*d^2*r^3 + 282*b*d^2*r^2 + 396*b*d^2*r
+ 216*b*d^2)*x^6*e*log(c) + (b*d^2*n*r^5 + 16*b*d^2*n*r^4 + 97*b*d^2*n*r^3 + 282*b*d^2*n*r^2 + 396*b*d^2*n*r +
 216*b*d^2*n)*x^6*e*log(x) + (a*d^2*r^5 - (b*d^2*n - 16*a*d^2)*r^4 - 36*b*d^2*n - (10*b*d^2*n - 97*a*d^2)*r^3
+ 216*a*d^2 - (37*b*d^2*n - 282*a*d^2)*r^2 - 12*(5*b*d^2*n - 33*a*d^2)*r)*x^6*e)*x^r)/(r^6 + 22*r^5 + 193*r^4
+ 864*r^3 + 2088*r^2 + 2592*r + 1296)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3655 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (137) = 274\).
time = 1.70, size = 1586, normalized size = 10.79 \begin {gather*} \frac {6 \, b d^{3} n r^{6} x^{6} \log \left (x\right ) + 108 \, b d^{2} n r^{5} x^{6} x^{r} e \log \left (x\right ) - b d^{3} n r^{6} x^{6} + 6 \, b d^{3} r^{6} x^{6} \log \left (c\right ) + 108 \, b d^{2} r^{5} x^{6} x^{r} e \log \left (c\right ) + 132 \, b d^{3} n r^{5} x^{6} \log \left (x\right ) + 54 \, b d n r^{5} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 1728 \, b d^{2} n r^{4} x^{6} x^{r} e \log \left (x\right ) - 22 \, b d^{3} n r^{5} x^{6} + 6 \, a d^{3} r^{6} x^{6} - 108 \, b d^{2} n r^{4} x^{6} x^{r} e + 108 \, a d^{2} r^{5} x^{6} x^{r} e + 132 \, b d^{3} r^{5} x^{6} \log \left (c\right ) + 54 \, b d r^{5} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 1728 \, b d^{2} r^{4} x^{6} x^{r} e \log \left (c\right ) + 1158 \, b d^{3} n r^{4} x^{6} \log \left (x\right ) + 12 \, b n r^{5} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 1026 \, b d n r^{4} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 10476 \, b d^{2} n r^{3} x^{6} x^{r} e \log \left (x\right ) - 193 \, b d^{3} n r^{4} x^{6} + 132 \, a d^{3} r^{5} x^{6} - 27 \, b d n r^{4} x^{6} x^{2 \, r} e^{2} + 54 \, a d r^{5} x^{6} x^{2 \, r} e^{2} - 1080 \, b d^{2} n r^{3} x^{6} x^{r} e + 1728 \, a d^{2} r^{4} x^{6} x^{r} e + 1158 \, b d^{3} r^{4} x^{6} \log \left (c\right ) + 12 \, b r^{5} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 1026 \, b d r^{4} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 10476 \, b d^{2} r^{3} x^{6} x^{r} e \log \left (c\right ) + 5184 \, b d^{3} n r^{3} x^{6} \log \left (x\right ) + 240 \, b n r^{4} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 7344 \, b d n r^{3} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 30456 \, b d^{2} n r^{2} x^{6} x^{r} e \log \left (x\right ) - 864 \, b d^{3} n r^{3} x^{6} + 1158 \, a d^{3} r^{4} x^{6} - 4 \, b n r^{4} x^{6} x^{3 \, r} e^{3} + 12 \, a r^{5} x^{6} x^{3 \, r} e^{3} - 432 \, b d n r^{3} x^{6} x^{2 \, r} e^{2} + 1026 \, a d r^{4} x^{6} x^{2 \, r} e^{2} - 3996 \, b d^{2} n r^{2} x^{6} x^{r} e + 10476 \, a d^{2} r^{3} x^{6} x^{r} e + 5184 \, b d^{3} r^{3} x^{6} \log \left (c\right ) + 240 \, b r^{4} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 7344 \, b d r^{3} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 30456 \, b d^{2} r^{2} x^{6} x^{r} e \log \left (c\right ) + 12528 \, b d^{3} n r^{2} x^{6} \log \left (x\right ) + 1836 \, b n r^{3} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 24624 \, b d n r^{2} x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 42768 \, b d^{2} n r x^{6} x^{r} e \log \left (x\right ) - 2088 \, b d^{3} n r^{2} x^{6} + 5184 \, a d^{3} r^{3} x^{6} - 72 \, b n r^{3} x^{6} x^{3 \, r} e^{3} + 240 \, a r^{4} x^{6} x^{3 \, r} e^{3} - 2376 \, b d n r^{2} x^{6} x^{2 \, r} e^{2} + 7344 \, a d r^{3} x^{6} x^{2 \, r} e^{2} - 6480 \, b d^{2} n r x^{6} x^{r} e + 30456 \, a d^{2} r^{2} x^{6} x^{r} e + 12528 \, b d^{3} r^{2} x^{6} \log \left (c\right ) + 1836 \, b r^{3} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 24624 \, b d r^{2} x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 42768 \, b d^{2} r x^{6} x^{r} e \log \left (c\right ) + 15552 \, b d^{3} n r x^{6} \log \left (x\right ) + 6696 \, b n r^{2} x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 38880 \, b d n r x^{6} x^{2 \, r} e^{2} \log \left (x\right ) + 23328 \, b d^{2} n x^{6} x^{r} e \log \left (x\right ) - 2592 \, b d^{3} n r x^{6} + 12528 \, a d^{3} r^{2} x^{6} - 468 \, b n r^{2} x^{6} x^{3 \, r} e^{3} + 1836 \, a r^{3} x^{6} x^{3 \, r} e^{3} - 5184 \, b d n r x^{6} x^{2 \, r} e^{2} + 24624 \, a d r^{2} x^{6} x^{2 \, r} e^{2} - 3888 \, b d^{2} n x^{6} x^{r} e + 42768 \, a d^{2} r x^{6} x^{r} e + 15552 \, b d^{3} r x^{6} \log \left (c\right ) + 6696 \, b r^{2} x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 38880 \, b d r x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 23328 \, b d^{2} x^{6} x^{r} e \log \left (c\right ) + 7776 \, b d^{3} n x^{6} \log \left (x\right ) + 11664 \, b n r x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 23328 \, b d n x^{6} x^{2 \, r} e^{2} \log \left (x\right ) - 1296 \, b d^{3} n x^{6} + 15552 \, a d^{3} r x^{6} - 1296 \, b n r x^{6} x^{3 \, r} e^{3} + 6696 \, a r^{2} x^{6} x^{3 \, r} e^{3} - 3888 \, b d n x^{6} x^{2 \, r} e^{2} + 38880 \, a d r x^{6} x^{2 \, r} e^{2} + 23328 \, a d^{2} x^{6} x^{r} e + 7776 \, b d^{3} x^{6} \log \left (c\right ) + 11664 \, b r x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 23328 \, b d x^{6} x^{2 \, r} e^{2} \log \left (c\right ) + 7776 \, b n x^{6} x^{3 \, r} e^{3} \log \left (x\right ) + 7776 \, a d^{3} x^{6} - 1296 \, b n x^{6} x^{3 \, r} e^{3} + 11664 \, a r x^{6} x^{3 \, r} e^{3} + 23328 \, a d x^{6} x^{2 \, r} e^{2} + 7776 \, b x^{6} x^{3 \, r} e^{3} \log \left (c\right ) + 7776 \, a x^{6} x^{3 \, r} e^{3}}{36 \, {\left (r^{6} + 22 \, r^{5} + 193 \, r^{4} + 864 \, r^{3} + 2088 \, r^{2} + 2592 \, r + 1296\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")

[Out]

1/36*(6*b*d^3*n*r^6*x^6*log(x) + 108*b*d^2*n*r^5*x^6*x^r*e*log(x) - b*d^3*n*r^6*x^6 + 6*b*d^3*r^6*x^6*log(c) +
 108*b*d^2*r^5*x^6*x^r*e*log(c) + 132*b*d^3*n*r^5*x^6*log(x) + 54*b*d*n*r^5*x^6*x^(2*r)*e^2*log(x) + 1728*b*d^
2*n*r^4*x^6*x^r*e*log(x) - 22*b*d^3*n*r^5*x^6 + 6*a*d^3*r^6*x^6 - 108*b*d^2*n*r^4*x^6*x^r*e + 108*a*d^2*r^5*x^
6*x^r*e + 132*b*d^3*r^5*x^6*log(c) + 54*b*d*r^5*x^6*x^(2*r)*e^2*log(c) + 1728*b*d^2*r^4*x^6*x^r*e*log(c) + 115
8*b*d^3*n*r^4*x^6*log(x) + 12*b*n*r^5*x^6*x^(3*r)*e^3*log(x) + 1026*b*d*n*r^4*x^6*x^(2*r)*e^2*log(x) + 10476*b
*d^2*n*r^3*x^6*x^r*e*log(x) - 193*b*d^3*n*r^4*x^6 + 132*a*d^3*r^5*x^6 - 27*b*d*n*r^4*x^6*x^(2*r)*e^2 + 54*a*d*
r^5*x^6*x^(2*r)*e^2 - 1080*b*d^2*n*r^3*x^6*x^r*e + 1728*a*d^2*r^4*x^6*x^r*e + 1158*b*d^3*r^4*x^6*log(c) + 12*b
*r^5*x^6*x^(3*r)*e^3*log(c) + 1026*b*d*r^4*x^6*x^(2*r)*e^2*log(c) + 10476*b*d^2*r^3*x^6*x^r*e*log(c) + 5184*b*
d^3*n*r^3*x^6*log(x) + 240*b*n*r^4*x^6*x^(3*r)*e^3*log(x) + 7344*b*d*n*r^3*x^6*x^(2*r)*e^2*log(x) + 30456*b*d^
2*n*r^2*x^6*x^r*e*log(x) - 864*b*d^3*n*r^3*x^6 + 1158*a*d^3*r^4*x^6 - 4*b*n*r^4*x^6*x^(3*r)*e^3 + 12*a*r^5*x^6
*x^(3*r)*e^3 - 432*b*d*n*r^3*x^6*x^(2*r)*e^2 + 1026*a*d*r^4*x^6*x^(2*r)*e^2 - 3996*b*d^2*n*r^2*x^6*x^r*e + 104
76*a*d^2*r^3*x^6*x^r*e + 5184*b*d^3*r^3*x^6*log(c) + 240*b*r^4*x^6*x^(3*r)*e^3*log(c) + 7344*b*d*r^3*x^6*x^(2*
r)*e^2*log(c) + 30456*b*d^2*r^2*x^6*x^r*e*log(c) + 12528*b*d^3*n*r^2*x^6*log(x) + 1836*b*n*r^3*x^6*x^(3*r)*e^3
*log(x) + 24624*b*d*n*r^2*x^6*x^(2*r)*e^2*log(x) + 42768*b*d^2*n*r*x^6*x^r*e*log(x) - 2088*b*d^3*n*r^2*x^6 + 5
184*a*d^3*r^3*x^6 - 72*b*n*r^3*x^6*x^(3*r)*e^3 + 240*a*r^4*x^6*x^(3*r)*e^3 - 2376*b*d*n*r^2*x^6*x^(2*r)*e^2 +
7344*a*d*r^3*x^6*x^(2*r)*e^2 - 6480*b*d^2*n*r*x^6*x^r*e + 30456*a*d^2*r^2*x^6*x^r*e + 12528*b*d^3*r^2*x^6*log(
c) + 1836*b*r^3*x^6*x^(3*r)*e^3*log(c) + 24624*b*d*r^2*x^6*x^(2*r)*e^2*log(c) + 42768*b*d^2*r*x^6*x^r*e*log(c)
 + 15552*b*d^3*n*r*x^6*log(x) + 6696*b*n*r^2*x^6*x^(3*r)*e^3*log(x) + 38880*b*d*n*r*x^6*x^(2*r)*e^2*log(x) + 2
3328*b*d^2*n*x^6*x^r*e*log(x) - 2592*b*d^3*n*r*x^6 + 12528*a*d^3*r^2*x^6 - 468*b*n*r^2*x^6*x^(3*r)*e^3 + 1836*
a*r^3*x^6*x^(3*r)*e^3 - 5184*b*d*n*r*x^6*x^(2*r)*e^2 + 24624*a*d*r^2*x^6*x^(2*r)*e^2 - 3888*b*d^2*n*x^6*x^r*e
+ 42768*a*d^2*r*x^6*x^r*e + 15552*b*d^3*r*x^6*log(c) + 6696*b*r^2*x^6*x^(3*r)*e^3*log(c) + 38880*b*d*r*x^6*x^(
2*r)*e^2*log(c) + 23328*b*d^2*x^6*x^r*e*log(c) + 7776*b*d^3*n*x^6*log(x) + 11664*b*n*r*x^6*x^(3*r)*e^3*log(x)
+ 23328*b*d*n*x^6*x^(2*r)*e^2*log(x) - 1296*b*d^3*n*x^6 + 15552*a*d^3*r*x^6 - 1296*b*n*r*x^6*x^(3*r)*e^3 + 669
6*a*r^2*x^6*x^(3*r)*e^3 - 3888*b*d*n*x^6*x^(2*r)*e^2 + 38880*a*d*r*x^6*x^(2*r)*e^2 + 23328*a*d^2*x^6*x^r*e + 7
776*b*d^3*x^6*log(c) + 11664*b*r*x^6*x^(3*r)*e^3*log(c) + 23328*b*d*x^6*x^(2*r)*e^2*log(c) + 7776*b*n*x^6*x^(3
*r)*e^3*log(x) + 7776*a*d^3*x^6 - 1296*b*n*x^6*x^(3*r)*e^3 + 11664*a*r*x^6*x^(3*r)*e^3 + 23328*a*d*x^6*x^(2*r)
*e^2 + 7776*b*x^6*x^(3*r)*e^3*log(c) + 7776*a*x^6*x^(3*r)*e^3)/(r^6 + 22*r^5 + 193*r^4 + 864*r^3 + 2088*r^2 +
2592*r + 1296)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d + e*x^r)^3*(a + b*log(c*x^n)),x)

[Out]

int(x^5*(d + e*x^r)^3*(a + b*log(c*x^n)), x)

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